## The errors

Unfortunately, we have discovered several errors in :

1. (i)

Lemma 5 in Sect. 4 is incorrect. A counterexample is given in Sect. 2 below.

2. (ii)

Theorem 5 in Sect. 5 is incorrect. It would be correct if we could replace $$t \ge -M_a$$ by $$t \ge 0$$ in the condition (39) in Theorem 4, but we are not free to do so, because the condition $$t \ge -M_a$$ is required by the increasing convex stochastic order used in Theorem 4.

3. (iii)

The presentation of Lemma 3 is incorrect, but this is fixable, as explained in Sect. 3.

4. (iv)

Proposition 1 is incorrect, but this is fixable. This proposition becomes correct if the condition $$g(0) = 0$$ is added, as holds in the intended Erlang example ($$E_k$$ for $$k \ge 2$$). The correction is needed because (57) in  is missing the term g(0)h(t).

These errors have serious implications. The error in Lemma 5 invalidates the proofs of Theorems 1 and 3. The error in Theorem 5 invalidates the proof of Theorem 2. Thus, Theorems 1–3 become conjectures remaining to be proved or disproved.

The error in the proof of Theorem 1 invalidates the proof of Theorem 8, which invalidates the proof of Theorem 7. However, we have obtained new results, which provide a new proof of Theorem 7, as explained in Sect. 4 below.

## Counterexample to Lemma 5

We will work with the two-point distributions as defined in Sect. 2.1 of . Assume that the mean is $$m = 1$$, the upper limit of the support is $$M = 5$$ and the squared coefficient of variation is $$c^2 = 1$$. Let $$X_0$$ and $$X_{u}$$ be random variables with the extremal two-point cdf’s $$F_0$$ and $$F_u$$, respectively. Then, $$P(X_0 = 2) = 1/2 = P(X_0 = 0)$$, while $$P(X_u = 5) = 1/17$$ and $$P(X_u = 3/4) = 16/17$$. It is known that $$X_0 \le _{3-cx} X_u$$, as stated in (34) of . Since $$E[X_0] = E[X_u] = 1$$ and $$E[X_0^2] = E[X_u^2] = 2$$, we also have $$X_0 \le _{2,2} X_u$$. However, contrary to Lemma 5 in , the ordering $$Y_0 \equiv (X_0 - 3/4)^{+} \le _{2,2} (X_u - 3/4)^{+} \equiv Y_u$$ fails to hold. This is easy to see, because $$Y_0$$ and $$Y_u$$ are the two-point distribution with $$P(Y_0 = 0) = 1/2 = P(Y_0 = 5/4)$$, while $$P(Y_u = 0) = 16/17$$ and $$P(Y_u = 17/4) = 1/17$$, so that we have a reverse ordering of the means: $$E[Y_0] = 5/8 > 1/4 = E[Y_u] = E[X_u] - 3/4$$. For the counterexample to the ordering under consideration, note that $$Y_0 + t \ge 0$$ and $$Y_u + t \ge 0$$ for all $$t \ge 0$$,

\begin{aligned} E[(Y_0 + t)^2]= & {} t^2 + 5t/4 + O(1) \quad \text{ and }\quad \\ E[(Y_u + t)^2]= & {} t^2 + t/2 + O(1) \quad \text{ as }\quad t \rightarrow \infty , \end{aligned}

so that $$E[(Y_0 + t)^2] > E[Y_u + t)^2]$$ for all t sufficiently large. This contradicts the claim of Lemma 5.

## Correcting Lemma 3

Lemma 3 is important because it provides a way to apply the theory of Tchebycheff (T) systems from , as briefly reviewed in  and Section 3 of . However, in the statement of Lemma 3 insufficient care was given to the support of the random variable Y with distribution $$\Gamma$$ appearing in (22) of . The support of Y should be chosen so that the integrand $$\phi (u)$$ appearing in (21) of  is not identically 0 for any subinterval of $$[0, M_a]$$. Hence, the support of Y should be changed from $$[0,\infty )$$ to a more general interval, i.e., (22) should be replaced by

\begin{aligned} \phi (u) \equiv \int _{a}^{b} h((y-u)^{+}) \, \mathrm{d}\Gamma (y) = h(0)\Gamma (u) + \int _{u+}^{b} h(y-u) \, \mathrm{d}\Gamma (y), \quad 0 \le u \le M_a, \end{aligned}
(1)

where

\begin{aligned} -\infty \le a \le 0 < M_a \le b \le \infty , \end{aligned}
(2)

$$\Gamma$$ is a cdf of a real-valued random variable Y with a continuous positive density function over the interval [ab]. Then, in Lemma 3 of  we should replace (25) by (2) above. The proof also needs to be adjusted accordingly. In particular, the revised proof is:

### Proof

First, observe that the finite mgf condition implies that all integrals are finite. In each case, we can apply Lemmas 1 and 2 of  with (1) and (2). To do so, we apply the Leibniz rule for differentiation of an integral with (1). Using (2), we have

\begin{aligned} \phi (u)= & {} \int _{a}^{b} h ((y-u)^{+}) \, \mathrm{d}\Gamma (y) = \int _{u}^{b} h (y-u) \, \mathrm{d}\Gamma (y) + h(0) \Gamma (u) \quad \text{ and }\quad \nonumber \\ \phi ^{(1)} (u)= & {} -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y) - h(0)\gamma (u) + h(0) \gamma (u)\nonumber \\&= -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y). \end{aligned}
(3)

For $$h(x) \equiv x$$ in condition (i), we have $$h^{(1)} (x) = 1$$ for all x, so that

\begin{aligned} \phi ^{(1)} (u) = -\int _{u}^{b} h^{(1)} (y-u) \, \mathrm{d}\Gamma (y) = -\int _{u}^{b} \, \mathrm{d}\Gamma (y) = -(1 - \Gamma (u)), \end{aligned}
(4)

so that, by the condition on $$\Gamma$$,

\begin{aligned} \phi ^{(2)} (u) = \gamma (u) > 0 \quad \text{ and }\quad \phi ^{(3)} (u) = \gamma ^{(1)} (u) < 0 \quad \text{ for }\quad 0 \le u \le M_a. \end{aligned}
(5)

From the form of $$\phi ^{(3)} (u)$$ in (5), we see that the condition on $$\gamma$$ is necessary as well as sufficient. We also see that the UB and LB are switched if instead $$\gamma ^{(1)} (u) > 0$$.

Turning to $$h(x) = x^2$$ in condition (ii), we use $$h^{(1)} (0) = 0$$ and $$h^{(2)} (x) = 2$$ for all x with the second line of (3) to get

\begin{aligned} \phi ^{(2)} (u) = \int _{u}^{b} h^{(2)} (y-u) \, \mathrm{d}\Gamma (y) = 2\int _{u}^{b} \, \mathrm{d}\Gamma (y) = 2 (1 - \Gamma (u)) > 0, \end{aligned}
(6)

so that $$\phi ^{(3)} (u) = -2 \gamma (u) < 0$$ for $$0 \le u \le M_a$$.

Conditions (iii) and (iv) are both special cases of condition (v), which implies that

\begin{aligned} \phi ^{(3)} (u) = -\int _{u}^{b} h^{(3)} (y-u) \, \mathrm{d}\Gamma (y) < 0.~~~ \end{aligned}
(7)

$$\square$$

## Application of Lemma 3 to the higher cumulants

In , we have applied the corrected Lemma 3 in  to develop new extremal results for the higher cumulants of the steady-state waiting time that provide corrected proofs of Theorems 7 and 8 in . These bounds for higher cumulants are interesting and important because they clearly demonstrate the value of Lemma 3 in  and highlight its limitation for treating the mean. In particular, the decreasing pdf condition in Lemma 3 (i) prevents positive results for the mean that we now obtain for the higher cumulants from Lemma 3 (ii) and (iii).